Finding the linear transformation of a matrix

In summary, we are given a matrix A2x2 with some random values and we are asked to say if there is a linear map which has A as its map for the standar basis. The Attempt at a Solution says that if A is the matrix of some linear map for the standar basis, then f(1,0)=(2,-5) and f(0,1)=(1,8).
  • #1
sphlanx
11
0

Homework Statement


Hello again. First of all thanks to anyone who has replied to my previous questions.
Now, the question that troubles me is:

We are given a matrix A2x2 with some random values and we are asked to say if there is a linear map which has A as its map for the standar basis.

A= 2 1
-5 8

Homework Equations





The Attempt at a Solution


If A is the matrix of some linear map for the standar basis that means that:

f(1,0)=2(1,0) -5(0,1)
f(0,1)=1(1,0) +8(0,1)

so f(1,0)=(2,-5) and f(0,1)=(1,8)

Although the question only asks to say if such a linear map exists (it obviously does but i don't know how to prove it) I would be glad if someone could instruct me on finding formulas for linear maps when we have their matrix.
 
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  • #2
So given a matrix, you are trying to find the linear transformation represented by the matrix. Building on the work you already have, and identifying the transformation by T, we have.

[tex]T\left(\begin{array}{c}x&y\end{array} \right)[/tex]
[tex]=~x~T\left(\begin{array}{c}1&0\end{array} \right)~+~y~T\left(\begin{array}{c}0&1\end{array} \right)[/tex]
[tex]=~x~\left(\begin{array}{c}2&-5\end{array} \right)~+~y~T\left(\begin{array}{c}1&8\end{array} \right)[/tex]
[tex]=~\left[\begin{array}{cc}?&?&?&?\end{array} \right]~\left(\begin{array}{c}x&y\end{array} \right)[/tex]

Can you fill in the missing entries in the 2 x 2 matrix?
 
  • #3
Mark44 said:
So given a matrix, you are trying to find the linear transformation represented by the matrix. Building on the work you already have, and identifying the transformation by T, we have.

[tex]T\left(\begin{array}{c}x&y\end{array} \right)[/tex]
[tex]=~x~T\left(\begin{array}{c}1&0\end{array} \right)~+~y~T\left(\begin{array}{c}0&1\end{array} \right)[/tex]
[tex]=~x~\left(\begin{array}{c}2&-5\end{array} \right)~+~y~T\left(\begin{array}{c}1&8\end{array} \right)[/tex]
[tex]=~\left[\begin{array}{cc}?&?&?&?\end{array} \right]~\left(\begin{array}{c}x&y\end{array} \right)[/tex]

Can you fill in the missing entries in the 2 x 2 matrix?

Yes I think I got it now. The missing matrix is the matrix i posted on my first post. So T(x,y)=(2x+y,-5x+8y). The way you put it, I see that a matrix and a linear transformation are actually the exact same thing, expressed in a different way? Thank you very much, the work you people do here is incredible.
 

Related to Finding the linear transformation of a matrix

1. What is a linear transformation?

A linear transformation is a mathematical function that maps a vector space onto itself in a way that preserves the algebraic structure of the space. This means that the transformation preserves operations such as addition and scalar multiplication.

2. How do you find the linear transformation of a matrix?

To find the linear transformation of a matrix, we first need to determine the transformation matrix. This can be done by applying the transformation to the standard basis vectors of the vector space and recording the resulting vectors. The resulting vectors will then form the columns of the transformation matrix.

3. What is the purpose of finding the linear transformation of a matrix?

The purpose of finding the linear transformation of a matrix is to understand how a given transformation affects a vector space. This can help us analyze and solve problems in various fields such as physics, engineering, and computer graphics.

4. Can a linear transformation have a determinant?

No, a linear transformation does not have a determinant. Determinants are only defined for square matrices, while linear transformations can be represented by matrices of any size. However, the determinant of the transformation matrix can give us information about the scale factor and orientation of the transformation.

5. How do you know if a transformation is linear?

A transformation is considered linear if it satisfies two properties: preservation of addition and preservation of scalar multiplication. This means that the transformation of the sum of two vectors must be equal to the sum of the individual transformations, and the transformation of a scalar multiple of a vector must be equal to the scalar multiple of the transformation of the vector.

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